Question

how to do laplace transform on a signal that is in frequency domain.

I have a signal that is in time domain and i used fft to plot it in the frequency domain, is there a way i could plot it in s domain. I know that i could do laplace transform directly from time domain to s domain, but I need to find out are these two results match each other.

Fs = 4; % samples per second

dt = 1/Fs; % seconds per sample

L = 10; % length of the signal

t = (0:dt:L-dt); % seconds

y = (2*exp((-0.05+3j)*t)+2*exp((-0.05-3j)*t)+...

5*exp((-0.1+2j)*t)+5*exp((-0.1-2j)*t)+...

7*exp((-0.01+10j)*t)+7*exp((-0.01-10j)*t));

figure

plot(t, real(y),'DisplayName','Real(y)');

hold on

plot(t, imag(y),'DisplayName','Img(y)')

hold off

lgd=legend;

title('Real part of Signal in Time response');

% Finding DFT of y(t)

% define the fft of y(t)

X = fftshift(fft(y));

% normalize the samples

X = 2*X / length(X);

% define the range of k, index

k = -(length(X)-1)/2:1:length(X)/2;

% Define the frequency based on the index as below

f = k/length(X) * Fs;

% now plot the magnitude spectrum of the signal

figure;plot(f,abs(X));grid;

xlabel('Frequency in Hz');

ylabel('Amplitude');

title('Magnitude spectrum of the signal y(t)');

Answer 1

clear;
clc;
close all;

Fs = 4; % samples per second
dt = 1/Fs; % seconds per sample
L = 10; % length of the signal
t = (0:dt:L-dt); % seconds
y = (2*exp((-0.05+3*1i)*t)+2*exp((-0.05-3*1i)*t)+5*exp((-0.1+2*1i)*t)+5*exp((-0.1-2*1i)*t)+7*exp((-0.01+10*1i)*t)+7*exp((-0.01-10*1i)*t));
figure
plot(t, real(y),'DisplayName','Real(y)');
title('Real part of Signal in Time response');
Y = fftshift(fft(y));
X = 2*Y / length(Y);
k = -(length(Y)-1)/2:1:length(X)/2;
f = k/length(Y) * Fs;
figure;plot(f,abs(X));grid;
xlabel('Frequency in Hz');
ylabel('Amplitude');
title('Magnitude spectrum of the signal y(t) using Fourier Transform');

% Now doing the laplace transform using symbolic variables
clear;
clc;
syms s t w;
y = (2*exp((-0.05+3*1i)*t)+2*exp((-0.05-3*1i)*t)+5*exp((-0.1+2*1i)*t)+5*exp((-0.1-2*1i)*t)+7*exp((-0.01+10*1i)*t)+7*exp((-0.01-10*1i)*t));
Y = laplace(y);
Y = subs(Y,s,1i*w); %replacing s = jw;
f_range = -2:0.1:2;
Y = subs(Y,w,2*pi*f_range);
figure;
plot(f_range,abs(Y));grid
xlabel('Frequency in Hz');
ylabel('Amplitude');
title('Magnitude spectrum of the signal y(t) using laplace Transform');

Please feel free to ask doubts in the comment section. Outputs are also attached below